Page 407 - Matrix Analysis & Applied Linear Algebra

5.11 Orthogonal Decomposition 403
5.11 ORTHOGONAL DECOMPOSITION

The orthogonal complement of a single vector x was defined on p. 322 to be the
set of all vectors orthogonal to x. Below is the natural extension of this idea.

Orthogonal Complement
Fora subset M of an inner-product space V, the orthogonal com-
⊥
plement M (pronounced “M perp”) of M is defined to be the set
of all vectors in V that are orthogonal to every vector in M. That is,

⊥
M = x ∈V m x =0 for all m ∈M .

2
For example, if M = {x} is a single vector in , then, as illustrated in
⊥
Figure 5.11.1, M is the line through the origin that is perpendicular to x. If
3
M is a plane through the origin in , then M ⊥ is the line through the origin
that is perpendicular to the plane.

Figure 5.11.1
⊥ ⊥
Notice that M is a subspace of V even if M is not a subspace because M is
closed with respect to vector addition and scalar multiplication (Exercise 5.11.4).
⊥
But if M is a subspace, then M and M decompose V as described below.
Orthogonal Complementary Subspaces

If M is a subspace of a finite-dimensional inner-product space V, then

⊥
V = M⊕M . (5.11.1)
Furthermore, if N is a subspace such that V = M⊕N and N⊥ M
(every vector in N is orthogonal to every vector in M ), then

⊥
N = M . (5.11.2)